2130_Chapter7_Notes

Lets look at an example x 1 1 x 1 3 find the

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Unformatted text preview: v2 = 0 1 −2 2 −4 1 −2 2 −4 0 v1 = v2 0 (first equation) =⇒ v1 = 2v2 =⇒ v = Check #2: (A − 2I)v = 3−2 −2 2 −2−2 2c 2 =c , c = 0. c 1 0 2 . = 0 1 9 / 12 5 Exa 2 (slide 3) λ2 = −1 =⇒ A − λ2 I = A + I = Check #1: det(A + I) = 3+1 −2 2 −2+1 4 −2 2 −1 4 −2 = −4 + 4 = 0. 2 −1 (A + I)v = 0 =⇒ 4 −2 2 −1 =⇒ 2v1 − v2 = 0 (second equation) =⇒ v2 = 2v1 =⇒ v = Check #2: (A + I)v = = 4 −2 2 −1 v1 0 = v2 0 c 1 =c , c = 0. 2c 2 1 0 = . 2 0 10 / 12 Exa 2 (slide 4) λ1 = 2, v(1) = λ2 = −1, v(2) = Fundamental matrix: Ψ(t) = 2 1 =⇒ x(1) = 1 2 =⇒ x(2) = 2 2t e 1 1 −t e 2 2e2t e−t . e2t 2e−t Next, compute the special fundamental matrix Φ(t) at t0 = 0. 11 / 12 6 Exa 2 (slide 4) Fundamental matrix: Ψ(t) = 2e2t e−t . e2t 2e−t Special fundamental matrix: Φ(t) = Ψ(t)Ψ−1 (0). Ψ−1 (0) 21 = 12 =⇒ Φ(t) = = Check: Φ(0) = −1 = 1 2 −1 2 3 −1 1 2e2t e−t 3 e2t 2e−t 2 −1 −1 2 1 4e2t − e−t 2(e−t − e2t ) 3 2(e2t − e−t ) 4e−t − e2t 1 4−1 2(1 −...
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